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Volume 11, Number 6

4 June 2010

Q06004, doi:10.1029/2010GC003038

ISSN: 1525‐2027

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Constraints on upper mantle viscosity from the flow‐inducedpressure gradient across the Australian continental keel

Christopher HarigDepartment of Geological Sciences, University of Colorado at Boulder, Benson Earth Sciences Building, Boulder,Colorado 80309, USA ([email protected])

Also at Cooperative Institute for Research in Environmental Science, University of Colorado at Boulder, Boulder,Colorado 80309, USA

Shijie ZhongDepartment of Physics, University of Colorado at Boulder, Duane Physics Building, Boulder, Colorado 80309, USA([email protected])

Frederik J. SimonsDepartment of Geosciences, Princeton University, Guyot Hall, Princeton, New Jersey 08544, USA([email protected])

[1] The thickness of continental lithosphere varies considerably from tectonically active to cratonic regions,where it can be as thick as 250–300 km. Embedded in the upper mantle like a ship, when driven to move bya velocity imposed at the surface, a continental keel is expected to induce a pressure gradient in the mantle.We hypothesize that the viscosity of the asthenosphere or the shear coupling between lower lithosphere andasthenosphere should control this pressure effect and thus the resulting dynamic topography. We performthree‐dimensional finite element calculations to examine the effects of forcing a continental keel by animposed surface velocity, with the Australian region as a case study. When the upper mantle is strongbut still weaker than the lower mantle, positive dynamic topography is created around the leading edge,and negative dynamic topography is created around the trailing edge of the keel, which is measurableby positive and negative geoid anomalies, respectively. For a weak upper mantle the effect is muchreduced. We analyze geoidal and gravity anomalies in the Australian region by spatiospectral localizationusing Slepian functions. The method allows us to remove a best fit estimate of the geographically localizedlow spherical harmonic degree contributions. Regional geoid anomalies thus filtered are on the order of±10 m across the Australian continent, with a spatial pattern similar to that predicted by the models.The comparison of modeled and observed geoid anomalies places constraints on mantle viscosity structure.Models with a two‐layer mantle cannot sufficiently constrain the ratio of viscosity between the upper andlower mantle. The addition of a third, weak, upper mantle layer, an asthenosphere, amplifies the effects ofkeels. Our three‐layer models, with lower mantle viscosity of 3 × 1022 Pa s, suggest that the upper mantle(asthenosphere) is 300 times weaker than the lower mantle, while the transition zone (400–670 km depths)has a viscosity varying between 1021 and 1022 Pa s.

Components: 13,900 words, 7 figures.

Keywords: asthenosphere; mantle; rheology; continental keel; geoid; spatiospectral localization.

Index Terms: 8122 Tectonophysics: Dynamics: gravity and tectonics (8033); 8162 Tectonophysics: Rheology: mantle(8033); 8103 Tectonophysics: Continental cratons.

Copyright 2010 by the American Geophysical Union 1 of 21

http://dx.doi.org/10.1029/2010GC003038

Received 11 January 2010; Revised 1 April 2010; Accepted 7 April 2010; Published 4 June 2010.

Harig, C., S. Zhong, and F. J. Simons (2010), Constraints on upper mantle viscosity from the flow‐induced pressure gradientacross the Australian continental keel, Geochem. Geophys. Geosyst., 11, Q06004, doi:10.1029/2010GC003038.

1. Introduction

[2] Our understanding of Earth’s deformation anddynamics fundamentally depends on the rheologyof the mantle. The viscosity structure of the mantlehas been inferred mainly by studying the responseto disappearing glacial surface loads over the past105 years [e.g., Cathles, 1975; Peltier, 1976;Wu andPeltier, 1983; Yuen and Sabadini, 1985; Nakada andLambeck, 1989; Lambeck et al., 1990; Mitrovica,1996; Simons and Hager, 1997; Mitrovica et al.,2007] and by examining geophysical signals frommodels of mantle convection, such as long‐wave-length geoid anomalies and surface plate velocities[e.g., Hager, 1984; Ricard et al., 1984; Hager andRichards, 1989; King and Masters, 1992; Forteand Peltier, 1994]. Results from these methods havenot always been consistent. Analyses of convection‐related observables have routinely suggested thatthe upper mantle is less viscous than the lowermantle by a factor of at least 30, and perhaps asmuch as 300. On the other hand, studies of glacialisostatic adjustment sometimes argue for less than afactor of 10 variation [Peltier, 1998]. Jointly invert-ing several types of data has provided additionaldetail [e.g., Mitrovica and Forte, 2004], but themantle’s viscosity structure remains incompletelyresolved. This is mainly due to the poor verticalresolution of the postglacial rebound data [Paulsonet al., 2007a, 2007b]. Here, we consider whetherpressure gradients across continental keels can beused to place a meaningful constraint on the vis-cosity of the upper mantle.

[3] The thickness of the continental lithosphere var-ies considerably from tectonically active to stablecratonic regions [Artemieva, 2009]. Determiningthe depths of continental keels has been an area ofmuch study and debate over the past several dec-ades [King, 2005], with estimates historically rang-ing from 175 to 400 km. Observations of surface heatflux, for example, suggest a thick Archean litho-sphere [e.g., Rudnick et al., 1998], though nowhereexceeding 250 km [e.g., Ballard and Pollack, 1987;Nyblade and Pollack, 1993; Jaupart et al., 1998].Analyses of mantle xenoliths, if indeed representa-tive of a conductive geotherm, have led to thicknessestimates in the lower end of the range, 150–200 km

[Rudnick et al., 1998]. Measurements of electricalconductivity show that differences between oceanicand Archean cratonic regions are limited to depthsshallower than 250 km [Hirth et al., 2000]. Seismi-cally, lithosphere is typically considered to extend todepths where shear wave speeds are significantlyfaster than the global average speed (usually >1.5%–2%) [Masters et al., 1996;Mégnin and Romanowicz,2000; Simons and van der Hilst, 2002; Ritsema etal., 2004]. While some types of data are known tobe influenced by anisotropy in the upper mantle[Ekström and Dziewoński, 1998;Gung et al., 2003],most recent seismic estimates generally limit litho-spheric thickness to at most 300–350 km [Artemieva,2009]. Overall, across disciplines, the continentalkeel thickness estimates are in the range of 200–300 km. In particular, Australia, our region of inter-est, consistently yields some of the highest esti-mates of any craton, with fast seismic wave speedanomalies persistent to depths of 250–300 km inmodels of VSV, the vertically polarized shear wavespeed [e.g., Debayle and Kennett, 2000a; Simons etal., 2002; Ritsema et al., 2004].

[4] The base of the lithosphere has much signifi-cance to geodynamics since, as a mechanical lowerboundary, it separates the rocks which remaincoherent parts of the lithosphere over geologictime from those below that are part of the con-vecting mantle [Turcotte and Oxburgh, 1967]. It isfor this reason that such thick continental keels areexpected to translate with plate motion over longtime scales; an observation that is corroborated bythe global correlation of continental crustal age withlithospheric thickness at long wavelengths [Simonsand van der Hilst, 2002]. Furthermore, continentalkeels influence the coupling between mantle andlithosphere, thus affecting net rotation of litho-sphere [Zhong, 2001; Becker, 2006, 2008] as wellas regional lithospheric deformation [Conrad andLithgow‐Bertelloni, 2006].

[5] The motion of continental keels through theupper mantle, which is relatively less viscous, canbe expected to induce pressure perturbations in themantle moving around them [Ricard et al., 1988].Such pressure gradients will mainly be controlledby the viscosity and thickness of the asthenosphericchannel below the lithosphere. If the viscosity of

GeochemistryGeophysicsGeosystems G3G3 HARIG ET AL.: CONTINENTAL KEEL MOTION AND THE GEOID 10.1029/2010GC003038

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this channel is low relative to the rest of the mantlethe pressure gradient should cause return flowbeneath the keel with little effect on dynamictopography at the Earth’s surface. However athigher asthenospheric viscosities the return flowshould be reduced in favor of a signal in the surfacetopography and hence gravity anomalies or thegeoid. Ricard et al. [1988] used an approximatemode‐coupling method to estimate these geoidalanomalies in the tens of meters.

[6] In this study we constrain the viscosity of theupper mantle by comparing modeled dynamic

geoid anomalies to observations. As these signals areproportional to the magnitude of velocity changeacross the mantle, we focus on the Australiancontinent, with its relatively large surface veloci-ties. We analyze the regional geoidal anomalies byspatiospectral localization using Slepian functions.Our results will also be applicable to understandingthe development of seismic anisotropy beneathcontinental cratons and the orientation of the lith-ospheric stress field surrounding them.

2. Analytical Treatment

[7] To illuminate the physics, we first consider asimplified problem in two dimensions (2‐D) thatcan be solved analytically by neglecting flow in thethird dimension, normal to surface motion. Weexamine the flow at two locations: in the far field,in which a lithosphere of uniform thickness movesover a layered viscosity structure, and the flowbeneath a keel, where a much thicker lithospheremoves over the same layers. In both of theselocations we would expect only horizontal flow.Therefore, from conservation of mass, the amountof horizontal flow at these locations should balanceeach other. This takes the form of the flux balance

ZKu zð Þdz ¼

ZFu zð Þdz; ð1Þ

where the material flux across a vertical plane isthe integral of the horizontal velocity function uover depth. Horizontal and vertical coordinates arerepresented by x and z, and the subscript K indicatesthe location under the keel, while F indicates thefar field (see the notation section). This is alsoillustrated in the cartoon in Figure 1a, where the

Figure 1. (a) Cartoon illustrating the mass balanceargument in the analytical treatment in section 2. Arrowsrepresent the amount of mass transported in each region.Since the lithosphere moves with constant motion, theflow in the mantle must balance the excess mass trans-ported in regions of thick lithosphere (i.e., the keelregion). (b) Dimensionless pressure gradients from thetwo‐layer analytical solution (equation (6)) for variouskeel thicknesses and g, the ratio of upper mantle to lowermantle viscosities. (c and d) Numerical experiment sche-matic. Figure 1c shows viscosity variation with depth.Solid line is the preferred model, and dashed line showskeel viscosity. Grey shades show variations of viscosityconsidered. Figure 1d shows assumed layering.Maximumkeel depth is 300 km. Upper mantle–transition zoneboundary is varied to set channel thickness between thekeel and transition zone.


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arrows representing the amount of mass flow changewith depth but sum to the same amount in eachregion. Since the motion of the lithosphere is con-stant, the flow in the mantle must balance the excessmass transported in regions of thick lithosphere(i.e., keel regions). In one‐dimensional channelflow [Turcotte and Schubert, 2002], the equation ofmotion can be written as

@�

@z¼ @p

@xor �

@2u

@z2¼ @p

@x; ð2Þ

where t is the shear stress and p the pressure. Theviscosity, h, is assumed to be constant in eachlayer. By integration we obtain an equation for thevelocity, u, as a function of depth, which is sub-sequently solved for by applying the boundaryconditions. These are: constant velocity at the sur-face, and fixed zero velocity at the bottom. In thefar‐field case of surface‐driven motion, ∂p/∂x canbe considered as zero, and we have linear velocityfunctions with velocity everywhere in the samedirection as the top surface. Underneath the keel, ahorizontal pressure gradient is allowed and can besolved for when balancing the material flux.

[8] We simplify our solutions by nondimensionaliz-ing pressure and coordinates by the relevant length,mass, and time scales. For layers with constant vis-cosity we use

p0 ¼ pu0�LMh0

� � and x0 ¼ x

h0; ð3Þ

where p′ is dimensionless pressure, x′ is dimension-less horizontal coordinate, u0 is the horizontalvelocity at the surface, hLM is the viscosity of thelower mantle, and h0 is the thickness of the uppermantle in the nonkeel region. Values such as thethickness of the upper mantle channel below thekeel, h, and the thickness of the lower mantle, d,nondimensionalize to h′ = h/h0 and d′ = d/h0. Wealso use k = h0 − h as the thickness differencebetween the keel and the surrounding lithosphere,written dimensionless as k′ = (h0 − h)/h0.

[9] For a uniformly viscous mantle the keel‐induced pressure gradient is a well‐known result,which varies as the cube of the channel thickness,h, written as

@p

@x¼ 6u0�

k

h3or

@p0

@x0¼ 6

k0

h03: ð4Þ

This is similar to Turcotte and Schubert [2002,equation 6‐22], except that we allow for a non-

zero far‐field flux equal to that of uniformly thicklithosphere.

[10] When the mantle has multiple viscous layers,the dependence on the thickness of the weakestlayer is more complex. For a two‐layered mantle,velocity is solved for in each layer, and then thedimensionless pressure gradient can be written as

@p0

@x0¼

�h02 � �d02

�d0 þ h0

� ��

1� �d02

�d0 þ 1

� �

1

2h02 � �d0� � � �2d0h0 � d02ð Þ � h02

�d0 þ h0

� �þ A

; ð5Þ

A ¼ � 1

3�d02 2d0 þ 3h0ð Þ þ h03

3: ð6Þ

Here, g = hUM/hLM is the ratio of the viscositiesof both layers. Setting the thickness of the lowermantle, d′, to zero reduces equation (6) to equation(4). Assuming that the thicknesses of the upper andlower mantle are fixed with the boundary at 670 kmdepth, we plot the pressure gradient versus g, theratio of upper mantle to lower mantle viscosities, forseveral keel thicknesses (Figure 1b). As expected, athicker continental keel results in larger dimen-sionless pressure gradients. More interesting, how-ever, is the variation with g. As g is decreased fromone (uniform viscosity mantle), pressure gradientsinitially increase even though upper mantle vis-cosity is lower. Pressure gradients eventually peak,and decrease with decreasing g.

[11] We also considered a three‐layered mantlewith a fixed 300 km thick keel and another divisionaround 400 km depth. While this system is morecomplex, the cases we checked showed a weakupper mantle may result in increased pressuregradients and thus increased dynamic topography,as in the two‐layered case.

[12] The analytical model illuminates the problemof flow‐generated dynamic topography in the fol-lowing ways. First, the addition of a weak layer inthe upper mantle can enhance the effect of conti-nental keels and increase pressure gradients. Sec-ond, the magnitude of surface velocity exerts strongcontrol over dynamic topography since it directlyscales the pressure gradient, as per equation (3):the higher the surface velocity, the more dynamictopography can be generated in the system.

[13] While idealized, a 2‐D analytical treatment ofthe problem easily illustrates our hypothesis: thatcontinental keels induce both horizontal variations


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in mantle velocity and pressure that are controlledby the details of the viscosity structure. We continuethis analysis with more realistic three‐dimensional(3‐D) calculations, focusing on the unique gravitysignals resulting from the dynamic topography. Theabsence or presence of these signals in Earth’sobserved gravity field then allow us to bound theplausible viscosity structure below.

3. Numerical Experiment Setup

[14] Our keel models are kept relatively simple,since we intend to examine the first‐order effects oftheir motion only. We begin by assuming bound-ary‐driven flow, and neglect mantle and crustalbuoyancy forces. This Stokes flow problem isgoverned by two of the conservation equationsof viscous fluids, those for mass and momentum,represented in dimensionless form as

rrrr � v ¼ 0; ð7Þ

�rrrrpþrrrr � � rrrrvþrrrrTv� �� ¼ 0; ð8Þ

where v, p, and h are the velocity vector, pressure,and viscosity, respectively.

[15] These equations are solved with the parallelfinite element code CitcomCU [Moresi and Gurnis,1996; Zhong, 2006]. We design our model space inregional spherical geometry to span depths fromthe surface to the core‐mantle boundary (CMB),and to cover a 120° by 120° area. Typical resolu-tion for each case is 192, 192, and 104 elements inlongitude, latitude, and radial direction, respec-tively, giving 0.625° per element of horizontalresolution. Vertical resolution is enhanced in theupper mantle at the expense of lower mantle res-olution to properly resolve the expected large ver-tical gradients in horizontal velocity. The uppermantle (between 70 and 670 km depth) has 10 kmper element of radial resolution, while the lowermantle and lithosphere above 70 km depth have57.8 km and 14 km per element of radial resolu-tion, respectively. We specify the thickness oflithosphere at every column of elements and centerthe keel in our model space at 60° longitude,0° latitude. The surface velocity is then fixed toresult from an Euler pole rotation with an axis at90° latitude with rotation magnitude of 1 cm/yr.We use a fixed boundary condition on the bottom,which will be discussed further later. On the sidesof our box that are parallel to the flow direction weuse reflecting boundary conditions. On the sidesperpendicular to flow, we use periodic boundary

conditions which allow free throughflow with iden-tical velocity on either side. Thus the combinedvelocity solutions for the side boundaries are masspreserving. Perturbations to the pressure field causedby the keel motion result in dynamic topographyat the surface. We analyze the gravity anomaliesassociated with this dynamic topography and makecomparisons to the observed field.

[16] Since dynamic topography directly scales withthe magnitude of surface velocity for the New-tonian rheology used in our calculations, we focusour study on the Australian continent, which is thefastest moving continental plate. When imposingsurface velocity we use the azimuth of plate motionat the center (130°E, 25°S) of Australia’s litho-spheric keel from the HS3‐NUVEL1A model,which is 1.81° east of north [Gripp and Gordon,2002] and, later, scale the results by the surfacevelocity magnitude of 8.267 cm/yr. The prescribedsurface boundary conditions are the driving force inour calculations and may do work on the calcula-tion medium [e.g., Han and Gurnis, 1999]. If thiswork induces significant stresses at the surface itmay influence the dynamic topography of ourcalculations. We performed calculations with var-ious lithospheric viscosities (including the keel)between 10 and 500 times that of the lower mantle.As long as the lithosphere is sufficiently moreviscous than the upper mantle, there was very littledifference in the resulting surface stresses, hencedynamic topography and geoid, indicating thatstresses at the surface are caused by the pressureperturbations in the upper mantle associated withkeel structure. Also, our use of periodic inflow/outflow boundary conditions likely minimizes thiseffect. While our calculations use surface motionsover a passive mantle, mantle flow beneath a fixedkeel could produce similar pressure gradients. Theimportant quantity is the net shear between thesurface and the underlying mantle which could beinfluenced by buoyancy‐driven flow, such as sub-duction. Accordingly we examined mantle flowbeneath Australia from a global mantle flow modeldriven by both plate motion and mantle buoyancy[see Zhang et al., 2010] to investigate whether thevelocity boundary conditions assumed at the topand bottom are valid. While this is discussed fur-ther in section 5.4, the results are broadly consistentwith what we assume in this regard.

[17] We use the upper mantle shear velocity tomog-raphy model CUB2.0 [Shapiro and Ritzwoller,2002] to create lithospheric keel thickness distribu-tions for our calculations (Figure 2a). Since theAustralian continent is surrounded by relatively


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young oceanic lithosphere [Müller et al., 2008],lithospheric thickness is set to a minimum of 70 km.At each increasing depth we use a +2% cutoffshear wave velocity perturbation contour withrespect to the ak135 reference model [Kennett etal., 1995] to estimate the extent of the continentalkeel. As mentioned earlier, estimates for thethickness of continental cratonic lithosphere fromseismic tomography depend on the type of dataused. The CUB2.0 model was created via a MonteCarlo inversion of global surface wave dispersiondata using both Rayleigh and Love waves. Shapiroand Ritzwoller [2002] specifically allow for radialanisotropy in their inversion, down to a depth of250 km. Where possible, we use the best fittingVSV estimate, which is consistently smaller thanthe VSH estimate. Below this depth their data isunable to constrain radial anisotropy and the esti-mated isotropic shear wave speed, VS, is used. Welimit the thickness of the lithosphere to 300 kmsince greater thicknesses are not supported by themajority of tomographic upper mantle models.

[18] Our models use layered viscosity structuresaccording to the schematic shown in Figures 1cand 1d. Mantle viscosities are constant with depthwithin each layer and are rendered dimensionlessby a reference value of 2 × 1021 Pa s. Lithosphericviscosity and keel viscosity, are set to a constantsignificantly higher than the mantle (e.g., 1024 Pa s).We begin with a two‐layered mantle with viscositycontrast at 670 km and vary the ratio of viscositiesof the lower and upper mantle. Calculations arethen performed with a three‐layered mantle withdivisions at 670 km and 400 km depth. Finally, ourexperiments also vary the thickness of the astheno-spheric channel from 50 to 150 km to examine thetrade‐off between channel viscosity and thickness.

4. Analysis of Gravity

[19] We seek to compare the dynamic geoid anoma-lies in our calculations to Earth’s observed geoidin an effort to constrain the viscosity structure ofthe upper mantle. This task is neither simple norstraightforward. The Earth’s gravitational potentialat a given surface point receives contributions fromthe mass distribution at all depths beneath andaround it. In the Australasian region (Figure 2b) wewould therefore expect the geoid [Lemoine et al.,1998] to reflect the mass redistribution processesthat occur in the surrounding subduction zones[McAdoo, 1981], and processes in the lower mantle[Hager and Richards, 1989], in addition to thedynamic signal that we must thus attempt to iso-

Figure 2. (a) Plot of keel depth from the tomographymodel CUB2.0 [Shapiro and Ritzwoller, 2002]. Wemap the +2% shear wave speed perturbation from initialmodel ak135 using VSV and set a maximum lithospheredepth of 300 km. (b) Colored EGM96 geoid heightwithout the degree l = 2 zonal spherical harmoniccoefficient. (c) Plot of the sum of squares

PNþ5�¼1 ga

2 ofthe first N + 5 eigenfunctions localized within a 30°circular region centered in western Australia for thebandwidth L = 0–8. The colored field shows the sensi-tivity of our filter to the region of interest. Overlain isthe 90% contour of this sensitivity.


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late. Fortunately, we expect our dynamic signalsto be localized both spatially and spectrally. Byexamining equivalently localized contributions toEarth’s gravitational potential field we can distin-guish possible dynamic signals from these otherregional contributions.

[20] The usual spherical harmonic representationof potential fields links spatial and spectral infor-mation through global spherical basis functionswhich have perfect frequency selectivity but nonein space [e.g., Freeden and Michel, 1999]. In orderto isolate a spatially localized contribution to thesignal, spectral and spatial concentration must bebalanced somehow.

[21] For instance, Simons and Hager [1997] devel-oped a procedure that constrains regional con-tributions to global spherical harmonic spectra toexamine the rebound of the Canadian shield afterremoval of its ice sheet. They constructed isotropicbandlimited windowing functions on domains withcircular symmetry from zonal spherical harmonics,according to a sensible but nonoptimal [see, e.g.,Wieczorek and Simons, 2005] concentration crite-rion. After their pioneering work, Simons et al.[2006] showed how to derive a family of opti-mally concentrated basis functions on domainswith arbitrarily irregular boundaries. As their con-struction uses all of the available spherical har-monics Ylm, of integer degree l = 0,…, L, and orderm = −l, …, l, the “Slepian” basis, ga, a = 1, …,(L + 1)2, as it has come to be known, is a perfectalternative to the spherical harmonics. Indeed, anyscalar geophysical function, s(r̂), that is bandlim-ited to degree L and lives (without loss of gener-ality) on the surface of the unit sphere can berepresented completely equivalently in either basis,

s r̂ð Þ ¼XLl¼0

Xl

m¼�l

slm Ylm r̂ð Þ ¼XLþ1ð Þ2

�¼1

s� g� r̂ð Þ: ð9Þ

[22] The Slepian functions, ga, which are band-limited to some degree L, are always constructedwith reference to a particular spatial region ofinterest, R, of area A, on the surface of the unitsphere, W. The criterion for concentration to theregion of interest is quadratic: the Slepian functionsare those that maximize their energy locally for theavailable bandwidth, following

� ¼

ZRg2� r̂ð Þd�Z

�

g2� r̂ð Þd�¼ maximum; ð10Þ

and where 1 > l > 0. Practically, they are given bythe spherical harmonic expansion

g� r̂ð Þ ¼XLl¼0

Xl

m¼�l

g� lmYlm r̂ð Þ; ð11Þ

where the coefficients, ga lm, are obtained by solvingthe eigenvalue equation

XLl0¼0

Xl0m0¼�l0

Dlm;l0m0gl0m0 ¼ � glm: ð12Þ

The four‐dimensional object whose elements Dlm,l′m′

are products of spherical harmonics, integratedover the region R, is called the localization “kernel”[Simons et al., 2006].

[23] The eigenvalues of this problem, l1, l2, …,l(L+1)2, sum to a space‐bandwidth product termedthe “spherical Shannon number,” N. Typically, Nis a good estimate of the number of significanteigenvalues, and thus of the number of well‐concentrated functions for the problem at hand. Asa result, an expansion of the signal in terms of itsfirst N Slepian functions provides a high‐qualityregional approximation to the signal in the region[Simons and Dahlen, 2006], at the bandwidthlevel L. Since

N ¼ Lþ 1ð Þ2 A4�

; ð13Þ

where A/(4p) is the fractional area of localization,the effective dimension of the Slepian basis ismuch reduced compared to the (L + 1)2 terms inthe spherical harmonic expansion. The Slepianfunctions are efficient for the study of geograph-ically localized geophysical signals, which aresparse in this basis [Simons et al., 2009],

s r̂ð Þ �XN�¼1

s� g� r̂ð Þ for r̂ 2 R: ð14Þ

[24] The geoid in the region of Australia (Figure 2b)is dominated by two striking features: a broad andlarge‐amplitude positive anomaly to the north nearIndonesia and the Western Pacific, and an equallybroad and large‐amplitude negative anomaly southof India trending to the southeast. Both anoma-lies are rather long‐wavelength features, and canbe attributed to the history of subduction and lowermantle structure in the area [e.g., Hager andRichards, 1989; Ricard et al., 1993]. A simpleestimate for the size of a dynamic keel‐related (i.e.,model‐generated) signal would be roughly the sizeof the keel itself. Therefore, we shall determine


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spatiospectrally localized functions to remove thelonger‐wavelength contributions to the regionalgeoid in and around Australia both in the observa-tions and in our model domain, thereby hopefullypreserving the signal. By removing the long‐wavelength geoid contributions and ascribing whatremains to the keel movement we run the risk ofincurring bias from an unexpected contribution tothe geoid in our analysis. However, the uniquespatial pattern in our modeled geoid and itsagreement with the filtered observed geoid supportsour modeling approach.

[25] We elect to use our functions to removebandwidths below L = 8. This roughly correspondsto the wavelength of our keel outline and ourexperience has shown that it acceptably balancesremoving broad, regional geoid features with thepreservation of sufficient model signal for analysis.We separate the observations into a low‐degree anda high‐degree part in both the spherical harmonicand the Slepian basis of bandwidth L, as

s r̂ð Þ ¼XLmax

l¼0

Xl

m¼�l

slm Ylm r̂ð Þ; ð15Þ

¼XLl¼0

Xl

m¼�l

slm Ylm r̂ð Þ þXLmax

l¼Lþ1

Xl

m¼�l

slm Ylm r̂ð Þ; ð16Þ

¼XLþ1ð Þ2

�¼1

s� g� r̂ð Þ þXLmax

l¼Lþ1

Xl

m¼�l

slm Ylm r̂ð Þ: ð17Þ

¼XN�¼1

s� g� r̂ð Þ þXLþ1ð Þ2

�¼Nþ1

s� g� r̂ð Þ

þXLmax

l¼Lþ1

Xl

m¼�l

slm Ylm r̂ð Þ: ð18Þ

Compared to the original expansion (15),equation (18) represents the signal with the low‐degree components separated into local (the firstterm) and nonlocal (the second term) contributions.The first term in equation (18) can thus be omittedin order to remove the local contributions to thelow‐degree signal.

[26] If we sum the squares of all of the Slepianfunctions the value N/A is reached everywhere onthe unit sphere [Simons et al., 2006]; by performingthe partial sum of the first N terms we obtain

XN�¼1

g2� r̂ð Þ � N

Afor r̂ 2 R: ð19Þ

By plotting the sum of the first several squaredSlepian eigenfunctions we can determine where thetruncated expansion is most sensitive and thus mostsuccessful at subtracting regional contributions. Wewill target our attention to the area where theanalysis reaches 90% of its maximum sensitivity bythis measure. In practice, this means that we shalltake the first N + 5 basis functions to guarantee theefficient removal of low‐degree signal from theregion of interest (Figure 2c).

[27] When the geoid is bandlimited to increas-ingly higher spherical harmonic degrees, shorter‐wavelength signals begin to dominate the field.Around Australia, the sharp density contrast betweenthe continental lithosphere and oceanic litho-sphere that is over 100 Ma old results in prominentgeoid anomalies along the coastline in the shorter‐wavelength geoid field. We apply a simple, approx-imate, correction for these anomalies, derived byHaxby and Turcotte [1978]. This correction assumesthat topography and bathymetry follow Airy iso-static compensation, and therefore it expresses thechange in the moment of the density distributionthat is expected when the thickness of crust varies.We apply this correction to the geoid from theEGM96 model [Lemoine et al., 1998] prior to theSlepian filtering technique.

[28] We illustrate the application of our method inFigure 3 using data from EGM96. The Slepianfunctions we use will be designed to fit the local-ized power at the low degrees of the geoid. Theyare bandlimited to L = 8 and are concentratedwithin a region of interest of colatitudinal radiusQ = 30° centered on colatitude �0 = 115° andlongitude = 130° (i.e., the center of the Austra-lian keel). The corresponding rounded Shannonnumber N = 5. Figures 3a–3c display various ver-sions of the EGM96 geoid height that are simplytruncated, namely, after removal of the degreesl through 2, through 8, and between 9 and 360,respectively, i.e.,

s1 ¼X360l¼3

Xl

m¼�l

slm Ylm; ð20Þ

s2 ¼X360l¼9

Xl

m¼�l

slm Ylm; and ð21Þ

s3 ¼X8l¼3

Xl

m¼�l

slm Ylm: ð22Þ

For reference we note that s1, in Figure 3a, is a fairapproximation to the Earth’s nonhydrostatic geoid.


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[29] In the course of our analysis we found thatreconstructing the low‐degree geoid with a Slepianbasis of more than N terms was necessary toobtain good fits to the modeled data. Therefore,Figures 2c and 3d–3f use N + 5 = 10 basis func-tions to remove the local signal. Including theseextra functions does not significantly affect thetrade‐off between spatial and spectral localization.In Figures 3d–3f we show the filtering process byfirst showing, in Figure 3d, the fit of the first N + 5Slepian eigenfunctions to the low‐degree EGM96geoid (i.e., s3 shown in Figure 3c), with the cir-cular region of concentration, the 90% contour ofsensitivity that is also shown in Figure 2c, out-lined in white. Figure 3e displays what remainsafter subtracting the Slepian fit from the low‐

degree EGM96 geoid, s3 shown in Figure 3c.Finally, Figure 3f shows the results of subtractingthe low‐degree Slepian fit (i.e., Figure 3d) from thefull EGM96 geoid shown in Figure 3a. In otherwords, we are plotting

s4 ¼X10�¼1

s� g�; ð23Þ

s5 ¼X81�¼11

s� g� ¼ s3 � s4; and ð24Þ

s6 ¼X81�¼11

s� g� þX360l¼9

Xl

m¼�l

slm Ylm ¼ s1 � s4: ð25Þ

Figure 3. Example of Slepian filtering technique for a low maximum bandwidth of L = 8. (a–c) Spectrally truncatedversions of the EGM96 geoid height. (d–f) The filtering process. Figure 3a shows the complete EGM96 geoid undu-lation with degree l = 2 removed. Figure 3b shows the geoid with all coefficients from l = 2 through l = 8 set to zero.Figure 3c shows the geoid between l = 3 and 8. In this example, our functions are designed to fit the localized powerof these low degrees. Figure 3d shows the fit of the first N+5 Slepian eigenfunctions to the low‐degree EGM96 geoid(Figure 3c), concentrated within a 30° circular region (outlined in white) centered over western Australia. Figure 3eshows the residual after subtracting the Slepian fit from the low‐degree EGM96 geoid. Overlain in white is the 90%contour of sensitivity from Figure 2c. Figure 3f shows the results of subtracting the low‐degree Slepian fit from thefull EGM96 geoid (Figure 3a). Figures 3b, 3e, and 3f are shown with the same color scale, as are Figures 3c and 3d.


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A comparison of s2 and s6 in Figures 3b and 3f,which are shown using the same color scale, showsthe changes due to the regional subtraction of low‐degree signal. These changes include some subtlechanges in geoid height such as slightly broaderand larger positive anomalies in northwesternAustralia, and broader negative anomalies alongthe southwestern coast of Australia. This compar-ison shows the difference between what would bethe “traditional” all‐spectral and the optimized“Slepian” spatiospectral approach to removing theregional low‐degree contributions to the geoid.

[30] In conclusion, spatiospectral filtering allows usto examine the geoid in and around Australiawithout being biased by power in the 0 → L degreerange that mostly arises from regions outside ofAustralia. The resulting geoid anomalies from thisanalysis in the Australian region show a distinctfeature with negative and positive anomalies ofamplitude of ∼10 m in the southern and northernparts of Australia, respectively (Figure 3f). Suchregional geoid anomalies are used in this study toconstrain mantle viscosity structure.

5. Three‐Dimensional NumericalResults

[31] We performed calculations using the keelshown in Figure 2a and varied the viscosity ofthe asthenosphere. From the vertical normal stressfield we calculate the dynamic topography at thesurface (Figure 4a) and at the core‐mantle bound-ary, which is then used to calculate surface geoidanomalies (Figure 4b). As upper mantle viscosity isvaried, the spatial patterns of topography and thegeoid remain roughly the same while the magni-tude fluctuates. At the leading edge of the keel,vertical normal stresses cause positive dynamictopography at the surface while near the trailingedge of the keel the reverse is true, with negativedynamic topography at the surface. The resultinggeoid anomaly is also positive at the leading edgeof the keel and negative at the trailing edge. Someasymmetry also results due to the shape of thelithospheric keel.

[32] While the example model outputs shown inFigures 4a and 4b have yet to be filtered, a similarpattern is seen to occur in the observed geoid afterlocally removing long wavelengths (Figure 3f).In both modeled and observed fields, broad pos-itive geoid anomalies occur along the coast ofnorthwestern Australia, and broad negative anoma-

lies along the southwestern coast. We explore thisbehavior further later, but this initial observationprovides context for some of our model results.

5.1. Two‐Layered Mantle

[33] The primary descriptor of our model results isthe magnitude of the dynamic geoid anomaly. In atwo‐layer mantle with a division at 670 km depth(Figure 4c), the anomaly magnitudes vary with thethickness and viscosity of the channel below thekeel. When the viscosity of the entire mantle isuniform (Figure 4c, dashed line), the channel belowthe keel is effectively very thick, and the magnitudeof the dynamic geoid anomaly (Figure 4c) is rela-tively small (contours indicate the maximum geoidanomalies reached). As upper mantle viscositydecreases, this channel effectively gets thinner asdeformation concentrates in the upper mantle, andthe geoid anomaly increases. This increase con-tinues until the viscosity contrast reaches approxi-mately 1:33 (third row of squares below dashedline). Eventually the low viscosity of the uppermantle is the dominant property, reducing stressand geoid magnitudes. This is similar to what weobserved from the simple analytical models plottedin Figure 1b.

5.2. Three‐Layered Mantle

[34] In a three‐layered mantle, the general resultsfrom two‐layer models remain valid. For a constantlower mantle viscosity (3 × 1022 Pa s in Figure 4d),when upper mantle (<400 km depth) viscosity isreduced relative to the transition zone (between400 and 670 km depth), geoid anomalies initiallyincrease as more flow is concentrated in the uppermantle. As upper mantle viscosity is decreasedfurther, the dynamic geoid anomalies are eventu-ally reduced as the low viscosity reduces stressmagnitudes. The effect of the weak channel is alsoapparent here more explicitly. In Figure 4d, thediagonal dashed line is for an upper mantle andtransition zone that have equal viscosities, whichcorresponds again to a two‐layer system withdivision at 670 km depth. Alternatively, the verticaldashed line denotes cases where the transition zoneand lower mantle are isoviscous. This represents atwo‐layer system with division at 400 km depth.Cases near the division at 400 km generally resultin larger anomaly magnitudes, except for tworegions: (1) when mantle viscosity is nearly uni-form (top right of Figure 4d) the channel is thickenough to dominate subtle changes in viscosity


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structure and (2) when the viscosity of the weakestlayer is low (bottom of Figure 4d), stress magni-tudes are low enough that changing channel thick-ness results in an insignificant change to anomalymagnitude.

5.3. Channel Flow

[35] Arguably, the controlling parameter of thisprocess is not the absolute depth of the continentalkeel, but the thickness of the asthenospheric chan-

Figure 4. Examples of model output. (a) Example dynamic topography at the surface for a calculation with hLM =3 × 1022 Pa s, hTZ = 3 × 1021 Pa s, and hUM = 3 × 1020 Pa s. The colored topography anomalies are scaled to asurface velocity of 1 cm/yr. The surface velocity vector is aligned with the plate motion vector at azimuth 1.81° fromnorth. In solid white we show the 90% contour of filter sensitivity from Figure 2c. In dashed white we show theoutline of the Australian keel (where thickness >100 km) determined from the CUB2.0 model. (b) Example dynamicgeoid anomalies at the surface from the same calculation, also scaled to 1 cm/yr of surface motion. (c–e) Contouredmagnitudes of unfiltered model geoid anomalies in m, scaled to the Australian surface motion of 8.267 cm/yr.Magnitude simply represents the difference between peak minimum and maximum anomaly (i.e., no patterninformation, about 3.6 m in Figure 4b). Hollow squares show model individual runs. Figure 4c shows mag-nitudes in a two‐layered mantle with division at 670 km depth. Figure 4d shows magnitudes in a three‐layeredmantle with divisions at 670 km and 400 km depth. Here the viscosity of the lower mantle is fixed at 3 × 1022 Pa s.Diagonal dashed line is where the upper mantle and transition zone have equal viscosity, equivalent to a two‐layered mantle divided at 670 km depth. Vertical dashed line is where the transition zone and lower mantle haveequal viscosity, equivalent to a two‐layered mantle divided at 400 km depth. (e) Contoured magnitudes ofunfiltered model geoid anomalies in m for different channel thicknesses. Lower mantle and transition zone vis-cosities are fixed at 3 × 1022 Pa s and 3 × 1021 Pa s, respectively. Horizontal dashed line shows where the uppermantle and transition zone are isoviscous.


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nel. Different keel thicknesses can combine withregional variations in the depths of upper mantle dis-continuities to change the thickness of the astheno-spheric channel below it [e.g., Gilbert et al., 2001].

[36] Accordingly, we performed calculations vary-ing the thickness and viscosity of this channel(Figure 4e). Once again, we contour the maximumresulting geoid anomalies. Channel thickness isvaried by adjusting the depth to the asthenosphere–transition zone viscosity contrast to examine caseswith 50 km, 100 km, and 150 km between thisboundary and the keel bottom, while keeping thekeel thickness at 300 km. Generally, an increasein channel thickness results in a smaller geoid

anomaly, as expected. This change is smaller thanone might expect, however, from a single‐layermodel (equation (4)), or even a two‐layered mantle(equation (6) and Figure 1), where pressure gra-dients are nonlinearly (e.g., cubically in uniformviscosity mantle models) related to channel thick-ness. Instead, in 3‐D it seems likely that this non-linearity is offset by flow that passes relativelyunconstrained around the sides of the keel region.

5.4. Flow Field

[37] As a continental keel moves through a lessviscous upper mantle we might expect the mantle

Figure 5. Mantle velocity at 200 km depth for two model cases. Vectors show horizontal velocity. Colors showvertical velocity, with positive values out of the page. The 10 cm/yr scale vector for horizontal motion is valid forFigures 5a–5c. Coastlines are outlined in white. In both cases, hLM = 3 × 1022 Pa s and hTZ = 3 × 1021 Pa s. Thedepths shown are at 200 km, and the black shape outlines the Australian continental lithosphere at this depth. (a) Acase with asthenospheric viscosity h = 3 × 1021 Pa s. (b) A case with asthenosphere viscosity h = 9 × 1018 Pa s. (c) Similarvelocity slice at 300 km depth from a global mantle flow model of Zhang et al. [2010]. Note the different scale forvertical velocity; magnitudes less than −5 cm/yr are black. (d) Vertical profile of velocity with depth for point inFigure 5c indicated by red dot (135°E, 25°S).


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to deform horizontally around the sides of the keel[Fouch et al., 2000] or vertically, beneath the keel.The largest expected control on this deformation isthe viscosity of the asthenosphere, and hence, inFigure 5, we show horizontal (vectors) and verticalvelocity (colors) at a depth in the asthenosphere fortwo of our cases, one with moderate and one with lowasthenosphere viscosity. Surprisingly, both cases arefairly similar.

[38] The case with moderate asthenospheric vis-cosity (h = 3 × 1020 Pa s) displays very little var-iation of velocity with latitude (Figure 5a). Thiscase also shows downgoing velocities at the lead-ing, and rising velocities at the trailing edge of thekeel, indicating that material predominantly flowsbeneath the keel rather than horizontally around it.

[39] While the case with lower asthenosphericviscosity (h = 9 × 1018 Pa s) does show variation inthe plane (Figure 5b), with velocity vectors deflectedaround the keel, these discrepancies do not exceedten degrees from the azimuth of surface motion.This second case also has vertical velocities that aresimilar to those in the first case, but with somewhatmore variability. It is not until asthenospheric vis-cosity is decreased even further that flow directionsstart to significantly deviate from the direction ofplate motion.

[40] Since flow in the mantle could affect the netflow across the keel, or perhaps vertically beneathit, we examine the velocity field beneath Australiafrom a global mantle convection model [Zhang etal., 2010, case FS1]. The model, shown for thepresent day in planform in Figure 5c and in profilein Figure 5d, is the result of a time‐dependentcalculation and includes both prescribed surfaceplate motion history and mantle buoyancy forces.The modeled horizontal motion in the mantle isbroadly consistent with our own model assump-tions: motion in the mantle, particularly in thehigh‐viscosity lower mantle, is low relative to themotion at the surface and is a good representationof the net shear across themantle. North of Australia,the vertical motion in the upper mantle is dominatedby the subduction zones that have velocity magni-tudes near −8 cm/yr. Under Australia there areseveral small‐scale downwellings below the litho-sphere, which are likely due to sublithosphericsmall‐scale convection aided by large plate motion[e.g., van Hunen et al., 2005]. Both types of ver-tical motion could influence the keel‐induced pres-

sure gradient, and should therefore be taken intoaccount when making interpretations.

6. Geoid Comparison WithObservations

[41] To constrain the upper mantle viscosity struc-ture we compare the dynamic geoid from our modelcalculations to the Earth’s observed geoid. Sinceour data are localized spatially as well as spectrally,and because Earth’s gravitational potential receivesmany different contributions across the spatial andspectral domains we apply the Slepian filteringtechnique discussed in section 4 to both model andobservations before comparing them.

6.1. Two‐Layered Mantle

[42] We begin by comparing geoids from our two‐layer models with the observed geoid. We calculatethe misfit by finding the 2‐D absolute value of theerror per measurement as

Misfit ¼ 1

n

Xni¼1

obsi �modelij j ð26Þ

where obsi is an observed geoid measurement at aspecific location, modeli is a model geoid mea-surement at the same location, and n is the totalnumber of values compared (which is identicalin every case studied). Misfit is calculated withina subregion that includes our largest‐amplitudemodel geoid anomalies (Figure 6a, dashed rectan-gle), excluding areas where model anomalies arelow. This area covers both continental and oceanicparts of the Australian region. If we were toexamine a null model, equation (26) will produce amisfit that represents the inherent power of theobserved field, approximately 3 m. For our cases,models with misfit values below this fit theobserved field better than a null model.

[43] When inspecting misfit for a two‐layeredmantle (Figure 7a), three scenarios emerge. First, amodel that produces minimal dynamic geoidanomalies, such as a uniform mantle of viscosity2 × 1022 Pa s, will produce a misfit around 3 m.Second, a model that reproduces the observed fieldresults in a minimum misfit. This can be seen inmodel case A in Figures 6b and 6c, whose residualsremain fairly uniform from north to south and havesmall amplitudes. Finally, a model that producesvery large geoid anomalies will overshoot the


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apparent signal in the observed field (model case Bin Figures 6d and 6e). While the average magnitudeof this misfit is of the same order as that from amodel with null signal, upon inspection it is clearthat the positive‐negative north‐south signature ofthe residual has reversed.

[44] The misfit for a two‐layered mantle reachesan absolute minimum when the lower mantle vis-cosity hLM = 5.3 × 1022 Pa s and the upper mantleviscosity is about 20 times smaller, hUM = 2.75 ×1021 Pa s. It is clear from Figure 7a, though, thatthere is a broad region with misfits near the min-imum where models can be considered acceptable.The trade‐off between effective channel thicknessand upper mantle viscosity implies that modelswith viscosity increases from the upper to thelower mantle with ratios between 3 and 300 couldbe considered supported by the data. In a two‐layered mantle, the viscosity jump between theupper and lower mantle cannot therefore be suffi-ciently constrained.

6.2. Three‐Layered Mantle

[45] In a three‐layered mantle, we fix lower mantleviscosity and plot how misfit varies for differentviscosities of the upper mantle and transition zone(Figures 7b and 7c). As described earlier, when aweak upper mantle layer is introduced, dynamicgeoid anomalies can increase. Here, this means thatwe should expect more variation in the pattern ofmisfit depending on the viscosity structure.

[46] When lower mantle viscosity is 2 × 1022 Pa s(Figure 7b), the dynamic geoid anomalies generallyhave low magnitudes. The minimum misfit sug-gests a structure that maximizes the dynamic geoidsignal. Thus a structure with an upper mantle vis-cosity of 1–2 × 1020 Pa s (about 100 times weakerthan the lower mantle) and a transition zone vis-cosity of 3–10 × 1021 Pa s is preferred.

[47] At lower mantle viscosities higher than 2 ×1022 Pa s, overall anomaly magnitudes increase, e.g.,to the values already shown in Figure 4d, and themisfit pattern becomes more intricate (Figure 7c).

Figure 6. Example model fits. Example cases are denoted by red squares in Figure 7. Model cases A and B are for atwo‐layered mantle, while case C is from a three‐layered mantle. Model cases are subtracted from the observed geoidfield within the dashed white box, yielding the plots of residuals. The dashed white box also marks the area used forcalculating misfit. All geoid fields are plotted using the same ±10 m scale. (a) Filtered observed geoid field.(b) Filtered model geoid from case A for a two‐layered mantle. (c) Residual for case A. (d) Filtered model geoid fromcase B for a two‐layered mantle. (e) Residual for case B. (f) Filtered model geoid from case C for a three‐layeredmantle. This example is similar to case A, and a residual is not shown.


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Regions of lowest misfit generally occur whenhUM < 1020 Pa s. For the cases shown where hLM =3 × 1022 Pa s, the absolute minimum misfitoccurs when hTZ = 2.7 × 1022 Pa s and hUM = 7.9 ×1019 Pa s, but the region of misfits near this

minimum is in fact quite broad. If we examineonly the cases with misfits <1.8 m (the darkestshades of blue in Figure 7c), we can make someinteresting further observations about the best fit-ting viscosity structure. In each of these cases there

Figure 7. Color‐shaded images of misfit between filtered model cases and observed geoid. Hollow squares identifymodel runs. The background observed geoid field has a mean power of about 3.1 m. Therefore, the misfit betweenthe observed field and a model with no (zero) geoid anomaly would be about 3.1 m. This occurs when upper mantleviscosity is very low (<1019 Pa s). Other instances of misfit about 3.1 m occur when model signal is roughly twicethe power (i.e., the model signal overshoots the observed signal, resulting in a residual with power equivalent to theoriginal observed field). (a) Model misfits for a two‐layered mantle with division at 670 km depth. Red squares Aand B denote cases shown in Figure 6. (b) Model misfits for a three‐layer mantle with lower mantle viscosity heldfixed at 2 × 1022 Pa s. (c) Model misfits for a three‐layer mantle with lower mantle viscosity fixed at 3 × 1022 Pa s.Red square denotes case C shown in Figure 6. (d) Model misfits for a three‐layered mantle for varying channelthicknesses. Channel thickness is determined by varying the depth to the upper mantle–transition zone viscositydiscontinuity. Lower mantle and transition zone viscosities are fixed at 3 × 1022 Pa s and 3 × 1021 Pa s, respectively.Dashed line indicates where the upper mantle and transition zone are isoviscous.


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is a factor of ∼300 between the viscosities of thelower and upper mantle. Meanwhile, the viscosityof the transition zone varies over an order ofmagnitude, indicating that it is less important oncethe upper mantle is sufficiently weak. We showsuch a model case with three layers that has a lowmisfit (<1.8 m, see Figure 7c) in Figure 6f (modelcase C). While the pattern of the model signalsremains fairly consistent between the two andthree‐layered cases, the changes in amplitude causethe best fits to shift to lower uppermantle viscosities.

[48] Finally, we examine the effects of the thicknessof the asthenospheric channel on the model geoidusing a set of calculations in which asthenosphericviscosity and channel thickness are varied whiletransition zone and lower mantle viscosities arefixed at 3 × 1021 Pa s and 3 × 1022 Pa s, respec-tively (Figure 7d). The misfit in these models is lesssensitive to asthenospheric channel thickness thanone might expect from the single‐layer analysis(equation (4)). This confirms the finding that lateralflow of mantle material around the sides of litho-spheric keels plays a role in the upper mantle.

7. Discussion

7.1. Constraints on Mantle ViscosityStructure

[49] Two classic methods to study mantle viscositymake use of observations associated with postgla-cial rebound, and long‐wavelength geoid anoma-lies. Generally, in studies of postglacial rebound,the Earth’s response to surface loads is modeledand fit to observations such as relative sea levelhistories [Peltier, 1976; Wu and Peltier, 1982;Mitrovica, 1996; Simons and Hager, 1997; Peltier,1998; Mitrovica et al., 2007] or time‐varying grav-ity anomalies from the Gravity Recovery and Cli-mate Experiment (GRACE) [Paulson et al., 2007a;Tamisiea et al., 2007]. In some long‐wavelengthstudies, geoid anomalies from the mantle’s internaldensity variations, which depend on the viscositystructure, are compared to the observed geoid [e.g.,Hager and Richards, 1989]. While studies of long‐wavelength geoid anomalies have suggested alower mantle that is significantly more viscous thanthe upper mantle [e.g., Hager, 1984; Ricard et al.,1984; Hager and Richards, 1989], the results ofpostglacial rebound studies are not always consis-tent among themselves, with some suggesting amore uniform mantle viscosity [e.g., Peltier, 1998],and others also arguing for a lower mantle that issignificantly stronger than the upper mantle [e.g.,

Lambeck et al., 1990; Han and Wahr, 1995; Simonsand Hager, 1997;Mitrovica and Forte, 2004]. Usingrelative sea level change and GRACE time‐varyinggravity data, Paulson et al. [2007a, 2007b] recentlyshowed that the inconsistency among the postgla-cial rebound studies owes to the poor depth reso-lution of the observations. In particular, Paulson etal. [2007a] showed that if the mantle is divided intotwo layers with division at 670 km depth, viscositymodels that have ∼5 × 1019 Pa s and ∼5 × 1022 Pa sfor the upper and lower mantle, respectively, pro-duce fits to both data types that are similar to thoseof a viscosity model with 5.3 × 1020 Pa s for theupper and 2.3 × 1021 Pa s for the lower mantle.

[50] The main objective of this study has been toseek additional constraints on upper mantle vis-cosity by examining the gravity anomalies causedby the pressure difference associated with movingAustralian continental lithosphere, a thick keelplowing through the mantle. Our study thereforerepresents a new method to constrain the viscositystructure of the mantle. We found that modeledgeoid anomalies caused by a moving continentallithosphere with a keel show remarkable similari-ties to the observations, with negative and positivegeoid anomalies in southern and northern Australia,respectively. Assuming that such geoid anomaliesare indeed caused by the pressure differenceinduced by the keel’s motion, we have shown thatgeoid anomalies, when properly filtered to accountfor localized, long‐wavelength effects, can provideuseful constraints on mantle viscosity.

[51] If the mantle is divided at 670 km depth intotwo layers, the geoid in Australia is best explainedby a mantle viscosity structure with 2.75 × 1021 Pa sand 5.3 × 1022 Pa s for the upper mantle and lowermantle, respectively, a factor of 20 increase.However, this viscosity structure does not appear tobe consistent with the relative sea level andGRACE data as shown by Paulson et al. [2007a].This difficulty can be resolved by introducing anadditional layer or weak asthenosphere from thebase of the lithosphere to 400 km depth. We foundthat such a weak asthenosphere tends to amplifythe effects of a continental keel. With our three‐layer models, and fixing lower mantle viscosity tovalues between 2 and 3 × 1022 Pa s, we found thatupper mantle viscosity (i.e., above 400 km depth)needs to be ∼1020 Pa s, or ∼300 times weaker thanthe lower mantle, in order to reproduce the geoidanomalies in Australia. Interestingly, this viscositystructure is generally permissible by the relativesea level and GRACE data, as shown by Paulsonet al. [2007a]. However, our result depends on


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the magnitude of lower mantle viscosity. If thelower mantle is too weak (1022 Pa s or less), thegeoid produced by the keel is too small to explainthe observations. Therefore, our study suggeststhat future geodynamics studies (e.g., on mantlestructure, heat transfer, and mantle mixing) andmantle rheology should consider the possibility ofrelatively high lower mantle viscosity of 2–3 ×1022 Pa s.

[52] Recently, Conrad and Behn [2010] used seis-mic anisotropy and lithospheric net rotation to con-strain model viscosities for the asthenosphere (downto 300 km depth in their models) and the transitionzone (between 300 km and 670 km depths) to be0.5–1 × 1020 Pa s and 0.5–1 × 1021 Pa s, respec-tively, while the lower mantle viscosity is fixed at5 × 1022 Pa s. Considering the difference betweenour models in dividing the viscosity layers, theviscosities for the upper mantle above 670 kmdepth from our study are quite similar to those ofConrad and Behn [2010]. However, these authorsdid not explore the dependence of their models onlower mantle viscosity.

7.2. Relevance to Other Continental Keels

[53] The first‐order controls on upper mantlepressure gradients in our calculations are the vis-cosity structure of the mantle and the magnitude ofsurface velocity. In addition to these primary con-trols, a set of secondary factors can influence thedynamic topography at the surface to a lesserextent. Our calculations are for a fixed keel size.If the length of the keel is increased (in the direc-tion of surface motion) then the distance betweenpositive and negative dynamic topography willincrease, and more power of the geoid anomalywill be at longer wavelengths. If the keel’s width isincreased (perpendicular to surface motion) thendynamic anomalies widen as well. In this instancemagnitudes of dynamic topography will also belarger since a wider keel displaces more mantle asit moves.

[54] One of the unique features of the Australiankeel is its asymmetry in the direction of surfacemotion (Figure 2a). From the thickest part of thelithosphere (at about 130° longitude) to the east,the lithosphere quickly thins, coincidentally withthe decrease in crustal age [Simons and van derHilst, 2002]. To the west this transition is moregradual and tends to follow the boundary betweencontinental and oceanic crust (along the westerncoast). The effect of this asymmetry is most easilyseen in the positive dynamic topography at the

leading edge of the keel (Figure 4a). Such a shapein other keels could result in unique dynamic geoidanomalies. A unique geoidal pattern would helpdistinguish pressure‐induced anomalies from otherprocesses that could be acting on keel edges suchas small‐scale or edge‐driven convection [e.g., Kingand Ritsema, 2000; Conrad et al., 2010].

[55] Several cratonic regions, such as NorthAmerica, western Africa, and Siberia [Artemieva,2009], have lithospheric keels as thick as Australia(>250 km). If keel‐induced pressure effects couldbe observed for these regions this could provideadditional constraints on mantle viscosity. Each ofthese regions has relatively slow surface motionthat could make it difficult, however, to detectdynamic signals as we did in our analysis. Weperformed our analysis for Australia because itslarge surface motion makes it the most likely toshow these effects. The keels in western Africa andSiberia have surface speeds below about 2 cm/yr[Gripp and Gordon, 2002], so to first order thegeoidal anomalies would have much reducedmagnitude. While also having low surface speedsof roughly 3 cm/yr, the North American keel couldstill have detectable anomalies due to its larger keelsize.

7.3. Seismic Anisotropy

[56] Viscous deformation in the upper mantle isdominated by the rheology of its most dominantmineral, olivine [Karato and Wu, 1993]. This defor-mation aligns elastically anisotropic olivine crystals[e.g., Verma, 1960] in a lattice‐preferred orienta-tion [McKenzie, 1979; Ribe, 1989] in the uppermantle, an effect that is regularly studied seismo-logically [e.g., Hess, 1964; Forsyth, 1975; Longand Silver, 2009]. Because of this relationship,observations of seismic anisotropy can be used toconstrain geodynamic models of mantle flow [e.g.,Conrad et al., 2007]. In practice, complexities suchas the strain history [e.g., Ribe, 1992], “frozen”lithospheric anisotropy [e.g., Silver, 1996; Savage,1999; Silver et al., 2001], the presence of water[Jung and Karato, 2001], grain boundary effects[e.g., Zhang andKarato, 1995], and so on,mean thatsuch constraints are fraught with uncertainty [e.g.,Savage, 1999; Kaminski and Ribe, 2001; Becker etal., 2006]. However, the first‐order approach ofinferring from the direction of seismic anisotropy thedirection of mantle flow has been fruitful, elucidat-ing, for example, patterns of flow around hot spotsor underneath oceanic plates [e.g., Becker et al.,2003; Behn et al., 2004; Walker et al., 2005].


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[57] Our own results for Australia can be brought tobear on this relationship, by examining ourinstantaneous flow field velocities in the context ofpublished regional studies of seismic anisotropy,recently summarized by Fouch and Rondenay[2006]. Generally, body wave measurements madeat seismic stations correlate with the large‐scalestructure at the surface, suggesting strong litho-spheric anisotropy [Fouch and Rondenay, 2006].Surface wave analyses, which provide better con-straints on the variation of anisotropy with depth,have been conducted throughout the Australiancontinent in the past decade [e.g., Debayle, 1999;Debayle and Kennett, 2000a, 2000b; Simons et al.,2002, 2003; Debayle et al., 2005]. Australia andNorth America have both been found to have sig-nificant (about 2%) azimuthal anisotropy at orbelow 200 km depth [e.g., Debayle et al., 2005;Marone and Romanowicz, 2007]. This deep anisot-ropy (below 150 km) mostly correlates with present‐day plate motion [e.g., Simons et al., 2002; Simonsand van der Hilst, 2003; Debayle et al., 2005].

[58] As mentioned earlier, we might expect mantleflow to deflect around a continental keel. Based onthe pattern of observed shear wave splitting mea-surements, Fouch et al. [2000] have suggested thisis the case in North America. At depths of 150 km,Debayle and Kennett [2000a] found that anisotropyin western and central Australia aligns with north‐south plate motion while eastern Australia displaysazimuthal anisotropy that appears to follow thecraton boundary. However, deformation akin tothat suggested by the anisotropy does not occurin our models unless asthenospheric viscosity isvery low (<9 × 1018 Pa s). Our results agree withthe large majority of anisotropy measurements atthese depths that align in the direction of surfaceplate motion [e.g., Debayle and Kennett, 2000b;Simons et al., 2003; Debayle et al., 2005], andsuggests that return flow occurring beneath thekeel is important.

8. Conclusions

[59] When continental keels are driven by imposedsurface motion, pressure perturbations cause posi-tive dynamic topography at the leading edge, andnegative dynamic topography around the trailingedge of the keel. Depending on the viscositystructure of the mantle, this dynamic topographycan be on the order of ±100 m and the corre-sponding geoid anomalies can be on the order of±10 m.

[60] When filtered to remove localized long‐wavelength anomalies using a technique developedusing Slepian functions, the Australian geoidclearly displays the expected pattern, with positiveand negative anomalies of about 10 m amplitudeat the leading and trailing edges of the craton,respectively. Our model results agree with the ob-servations: assuming that the signal is indeedcaused by the dynamic motion of the keel, we areable to obtain constraints on the mantle viscositystructure below the continent.

[61] Dynamic topography produced by motion ofa continental keel depends strongly on the effec-tive thickness and viscosity of the asthenosphere,where most of the horizontal motion occurs. For atwo‐layered mantle with a division at 670 km,decreasing upper mantle viscosity can increasedynamic topography if viscosities are large enough.The minimum misfit between the modeled andobserved geoid occurs when hLM = 5.3 × 1022 Pa sand hUM = 2.75 × 1021 Pa s. However, these vis-cosities appear too large compared with postglacialrebound studies. This suggests that radial mantleviscosity variations are not fully captured by two‐layer models.

[62] In a three‐layer mantle, misfit patterns becomemore complex as lower mantle viscosity isincreased. For a lower mantle viscosity hLM = 3 ×1022 Pa s, the minimum misfit occurs when theupper mantle viscosity hUM = 7–10 × 1019 Pa s, afactor of about 300 smaller than that of the lowermantle, while the transition zone viscosity hTZ mayvary between 1021 and 1022 Pa s. Such a viscositystructure is not inconsistent with postglacial reboundstudies. Since our results are sensitive to lowermantle viscosity, they also suggest that a relativelyhigh lower mantle viscosity should be consideredin future geodynamic studies.

Notation

A area of geographical region of interest.Dlm,l′m′ spatiospectral localization kernel.

d thickness of the lower mantle.d′ dimensionless thickness of the lower

mantle.F location far from keel for analytical

treatment.ga Slepian basis function on the unit sphere.

ga lm spherical harmonic coefficients of theSlepian function ga.

h0 thickness of the upper mantle.h thickness of the low‐viscosity channel.


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h′ dimensionless thickness of the low‐viscosity channel.

K location under keel for analyticaltreatment.

k thickness difference between keel andsurrounding lithosphere (h0 − h).

k′ dimensionless thickness difference betweenkeel and surrounding lithosphere.

L degree of bandlimit of geophysical signal.Lmax maximum degree of expansion of

geophysical signal.l degree of spherical harmonic.m order of spherical harmonic.N spherical Shannon number.n number of observations compared in

misfit calculation.p pressure.p′ dimensionless pressure.R spatial region of interest.s scalar geophysical function on the unit

sphere.slm spherical harmonic coefficients of the

function s.sa Slepian basis coefficients of the function s.u horizontal component of velocity.v velocity vector.x horizontal coordinate.x′ dimensionless horizontal coordinate.

Ylm spherical harmonic on the unit sphere.z vertical coordinate.h viscosity (Newtonian).

hUM viscosity of the upper mantle.hTZ viscosity of the transition zone.hLM viscosity of the lower mantle.g ratio of viscosities between mantle layers.l Slepian eigenvalues, or the fraction of sig-

nal energy concentrated locally.W unit sphere. longitude.t shear stress.Q colatitudinal radius of the region of interest.�0 colatitude.

Acknowledgments

[63] This research was supported by the David and LucilePackard Foundation and the National Science Foundation.We thank Nan Zhang for providing the data presented inFigures 5c and 5d. Helpful reviews by Clint Conrad andScott King improved the clarity and quality of this work.Figures 1–7 were created with the GMT software [Wessel and

Smith, 1998]. The Slepian analysis routines are available athttp://www.frederik.net.

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